m 


lAVvTl-XX* 


UL  23  1913 

HE  ASYMPTOTIC  DEVELOPMENT 


FOR    A    CERTAIN 


INTEGRAL  FUNCTION  OF 
ZERO  ORDER 


Of   n{£ 
UNIVfiLiiS«T> 


BY 


.   CHARLES   W.   COBB 

ASSISTANT    PROFESSOR    OF    MATHEMATICS 
IN    AMHERST    COLLEGE 


PORTION     OF     A     THESIS     PRESENTED     TO     THE     FACULTY     OF     THE 
GRADUATE    SCHOOL    OF    THE    UNIVERSITY    OF    MICHIGAN 
IN    CANDIDACY    FOR    THE     DEGREE     OF 
DOCTOR    OF    PHILOSOPHY 


PRINTED    AT 

K\jt  Nortoootr  ^re^g; 

NORWOOD,  MASS. 

1913 


THE  ASYMPTOTIC  DEVELOPMENT 

FOR    A    CERTAIN 

INTEGRAL  FUNCTION  OF 
ZERO  ORDER 


BY 

CHARLES  W.   COBB 

ASSISTANT    PROFESSOR    OF    MATHEMATICS 
IN   AMHERST    COLLEGE 


A     PORTION    OF     A     THESIS     PRESENTED     TO     THE     FACULTY     OF     THE 

GRADUATE    SCHOOL    OF    THE    UNIVERSITY    OF    MICHIGAN 

IN    CANDIDACY    FOR    THE     DEGREE    OF 

DOCTOR    OF    PHILOSOPHY 


PRINTED   AT  , 

K\it  Nortoociti  Press 

NORWOOD,  MASS. 
1913 


%' 


BIBLIOGRAPHY 

Barnes,  E.  W.  A  Memoir  on  Integral  Functions.  Phil.  Trans,  {k), 
vol.199.     (1902). 

The  Classification  of  Integral  Functions.     Camb.  Phil.  Trans., 

vol.  19. 

The  Maclaurin  Sum  Formula.     Proc.  Lond.  Math.  Soc,  Series 

2,  vol.  3. 

The  Asymptotic  Expansion  of    Inteo-ral  Functions  of  Finite 

Non-zero  Order.     Proc.  Land.  Math.  Soc,  Series  2,  vol.  3. 

Ford,  W.  B,  On  the  Determination  of  the  Asymptotic  Develop- 
ments of  a  given  Function.  Annals  of  Math.,  2d  Series,  vol.  11 
(1910). 

Hardy,  G.  H.  On  the  Function  Pp(x).  Quarterly  Journal,  vol.  37 
(190.5). 

LiTTLEWOOD,  J.  E.  On  the  Asymptotic  Approximation  to  Integral 
Functions  of  Zero  Order.  Proc.  Lond.  Math.  Soc,  Series  2,  vol.  5 
(1907). 

On  the  Dirichlet  Series,  and  Asymptotic  Expansions  of  Inte- 
gral Functions  of  Zero  Order.  Pi^oc.  Lond.  Math.  Soc,  Series  2, 
vol.  7  (1909). 

On  a  Class  of  Integral  Functions.     Trans.  Camb.  Phil.  Soc, 

vol.21  (1910). 

jNIattson,  R.  Contributions  a  la  Th^orie  des  Fonctions  entieres. 
(These)  Upsal,  1905. 

INIellin,  H.     Om   definita  integraler  etc.      Acta  Soc.   Fenn.,  t.   20, 

No.  7. 

Uber  eine  Verallgemeimerung   der    Riemannschen    Function 

I  {s).     Acta  Soc  Fenn.,  t.  24,  No.  10. 

268473 


'2  ASYxMPTOTlC   DEVELOPMENT 

M ELLIN,  H.  Ein  Forinel  fiir  den  Logarithnms  transcendenter  Funk- 
tionen  von  endlichem  Geschlecht.  Acta  Soc.  Fenn.,  t.  29 
(1900).     Reprinted  Acta  Math.,  vol.  28  (1904). 

Peterson,  J.     Vorlesungen   iiber   Functionentheorie.     Copenhagen, 

1898. 

PoixcARE,  H.  Siir  les  int^grales  irregulieres  des  Equations  lin^aires. 
Acta  Math.,  vol.  8  (1886). 

1.  Introduction.  By  means  of  the  Maclaurin  Sum 
Formula,  Barnes  *  and  Ford  f  have  obtained  asymptotic 
developments  for  certain  integral  functions  of  non-zero 
order,  but  for  functions  of  zero  order  Barnes'  analysis 
breaks  down,  while  Ford  has  not  treated  the  case,  and 
Littlewood  J  holds  that  these  functions  should  be  studied 
by  special  methods.  The  present  paper,  however,  uses 
the  Maclaurin  Sam  Formula  to  obtain  the  asymptotic 
development  for  a  typical  function  of  zero  order,  and  may 
therefore  be  considered  as  supplementing  the  work  of 
Barnes  and  Ford,  by  showing  that  a  common  method  of 
investigation,  based  on  the  Maclaurin  formula,  may  be 
employed  in  determining  asymptotic  developments  for 
integral  functions  of  both  orders. 

2.  The  Maclaurin  Sum  Formula.  The  Maclaurin  Sum 
Formula  with  remainder  will  be  used  in  the  following 
form  :  || 

\if(w')  is  analytic  throughout  a  vertical  strip  of  the  w 
complex  plane,  extending  to   an  infinite  distance  above 

*Bakxes,  E.  W.  p.  Phil.  Trans.  {A)  vol.  199,  pp.  411-500;  Proc. 
Lond.  Math.  Soc,  Series  2,  vol.  8,  pp.  27.3-295. 

t  Ford,   W.  B.     Annals  of  Math.,  second  series,  vol.  11  (1910). 

X  LiTTLEWooi),  J.  E.     Proc.  Lond.  Math.  Soc,  Series  2,  vol.  6  (1907). 

II  Ford,  VV.  B.  Lectures  on  Divergent  Series  given  at  Univ.  of 
Michigan,  1910-1911. 


ASYMPTOTIC   DEVELOPMENT  3 

and  bslow  the  axis  of  reals  and  including  the  real  points 
w  =a^w  —  h^  and  is  such  that 

where  y  is  some   assignable   positive    quantity,  we    may 
write 

+  ^"P""!^"  [/'^"'-"(S)  -/'^'"-"ra)]  +  E„, 
(^  m)l 

where 

^--(2^;OJ^.Jo ^ ^   ^     ^ 

^  =  1  when  m  =  0, 

0  <  6*  <  1  whenm  =  l,  2,  3.... 

3.  Problem.  We  shall  apply  this  theorem  to  the  dis- 
cussion of  a  single  type-function  of  zero  order,  though 
the  m.ethod  employed  is  evidently  capable  of  broader 
application. 

The  problem  is  as  follows : 

Given  the  integral  function  of  zero  order, 

F(^z^  =  TT  M j  =  exp.  ^(2) ;  z  real  or  complex, 

it  is  proposed  to  consider  the  existence  and  determina- 
tion   of   an   asymptotic    development    for    H(z^    in    the 


^w=/(0+K^) 


4  ASYMPTOTIC   DEVELOPMENT 

precise  sense   of  "asymptotic"  as   originally  formulated 
by  Poincare,*  viz.,  a  development  of  the  form 

lime(2)=0;  w  =  0,  1,  2  .... 

This  problem  has  been  considered  by  other  methods  by 
Mellin,f  Barnes, J  Hardy,§  Mattson,||  and  Littlewood.^ 

So  far  as  results  are  concerned,  all  agree  as  to  the  lead- 
ing terms  (terms  that  go  to  infinity  as  z  becomes  infinite), 
but  no  two  results  are  exactly  alike  for  the  rest  of  the 
development.  Such  differences  as  present  themselves  are 
doubtless  of  form  only  and  hence  of  minor  importance,  but 
the  present  treatment  by  means  of  the  Maclaurin  Sum 
Formula  tends  to  unify  the  problem  of  determining  asymp- 
totic developments  for  integral  functions,  by  reducing 
cases  of  both  non-zero  and  zero  order  to  a  common  method 
of  treatment. 

4.    From  our  definition,  §  3, 
(1)  iT  (2!)  =  lim  r  V  log  (e^  -z)-VJ 

and  by  the  logarithm  of   a  complex  number  Z,  say,  we 
shall  mean  :   log  Z  —  log  R  +  i^  ; 

Z  =  B(gos  <t> -{- 1  sin  cp)  ;    —  2  7r<<I><0. 

In  order  to  evaluate  ff(z)  for  large  z,  we  first  regard 

z    as    fixed    and    place    —=  6 ;    p  =  greatest   integer   in 

z 

*  Acta  Math.,  vol.  8  (1886),  pp.  295-344. 

t  Acta  Soc.  Fennicce,  t.  XXIX.   §  Quarterly  Journal  of  Math.  (1905). 

I  Phil  Trans.  {A)  vol.  199.         ||  These,  Upsal  (1905). 

IT  Proc.  Lond.  Math.  Soc,  Series  2,  vol.  5  (1907). 


ASYMPTOTIC   DEVELOPMENT 


log  \z\;  -<  I  ^  I  <  1.     Inasmuch  as  the  following  analysis 

holds  only  when  -<  \d\<\,z  may  not  lie  on  any  circle 

whose  center  is  at  the  origin  and  whose  radius  is  e'' 
(p  =  1,  2,  3  •••)  but  z  may  be  chosen  anywhere  else  in  the 
complex  plane. 

b-h 

To  evaluate  V  /(a:),  the  theorem  of  §  2  requires  that 

the  corresponding  f  {vf)  shall  be  analytic  throughout  a 
vertical  strip  of  the  w-plane,  including  the  points  w  —  a^ 
w  =  h.     So  we  write 


:r-l 


p-\ 


(2)   ^  log  {e-  -  2)  =  ^  log  (g-  -  2)  -f  log  (6^  -  z) 

x=l  a-=l 

and  observe  that  the  function  f(w')  =  log  (e**  —  z')  is  an- 
alytic in  the  vertical  strip  containing  the  points  ?^  =  1, 
w  =p^  and  also  in  the  strip  containing  the  points  w  —p  + 1, 
w  =  X. 

We  have  then,  putting  w  =  0,  ^  =  1, 

V  log  (e^  -z)=    P  log  (g^  -  2)t7a:  -  |  [log  (e^  -  z) 

-iog(.-^)]  +  xi,(p)-n,(i) 

where 

fi,  (a:)  =  -  *  j^*  [log  (e^-^^-^  -  2)  -  log  (.-'^^  -  O]  .  ^^. 
Also, 


y  log  (e^  -  2)  =  f  log  (e^  -z)dx-\  [log  (g^  -  3) 
-  log  (e^+^  -  2)]  +  fl,(:r)  -n,(p-\'  1). 


ar=30+l 


6 


ASYMPTOTIC   DEVELOPMENT 


Taking  the  terms  of  the  right-hand  members  in  the 
above  order, 


r  log  (e^-z)  dx  =  [2:  log  {e^-z)-^-  f-^ 

=:p\og(e^-^)-^-\og(e-z)-^l-   T-^ 

Now, 

CP    xz      ,         C^    xz      n 

—  I    dx  =  I dx 

•^1   e^  —  z  *^i    z  —  e-^ 


dx 


dx. 


X"' 


z      \z 


The  power  series  in  —  is  uniformly  convergent  for  all 

x^  1  -^  X  ^  p^  since  in  this   interval  |  g*  |  <  |  2  [.     Hence, 
integrating  term  by  term, 

--  I    dx 

^1   e'^  —  z 

=  g  +  J(._l)H.1gJ(2.-l):,lgJ(3.-l)+...] 
where 

Collecting  terms, 
(3)    log  (^p  -  2)  +  ^  log  (gx  _  2j)  =  (^  +  1)  log  (eP  -  2) 

.r=l 

-  log  Ce-z)  +  s\  +  .sy  +  njp)  -  n,(i). 


ASYMPTOTIC   DEVELOPMENT 
Consider  next  the  terms  arising  from 

x-l 


We  have 


x-l 

X 

x=p+l 


log  (e=^  —  z)dx  =  \  X  k)g  (^^  -  ^)  —  -;t  —   I  — *^^ 


^^'i-;hi'- 


dx\ 


Now 

\^\<\f\ 

•-  X=IJ  +  \ 

if  we  neglect  terms  that  go  to  zero  exponentially  when  x 


is   infinite.       Calling  the  power  series  in 
may  write 


>p+i 


aS'o,  we 


x-l 

X 

X=p+1 


x'^ 


(4)    ]^log(e^-2)  =  a;log(g^-^)--2 -(i?H-l)log(6!^+i-2!) 


^(P  +  '^y_hogCe^-z)  +  hog(eP^-^-z)  +  S,^-hn,(x-) 


x-l 


Noting  that  V  a:  =  — - —  and  that 


(2:--jlog  (e^-2)-| 


^2      a^  —  X 


2 


=  0,  we  have 


(5)  II(z)  =  (p  +  1)  log  (e^  -z)-  I  log  (e  -  z^ 

+  ^1  +  s^  +  aS'^  +  n,(2;)  -  n,  (i?  +  1) 
+  n,(jt>)-n,(i). 


8  ASYMPTOTIC   DEVELOPMENT 

To  evaluate  jS^  and  xS'g,  note  that  since  each  series  is  ab- 
solutely convergent,  we  may  write,  setting 


for,  loga-'-)^_i_l-_rL^_... 

r  2      3 


\eej      AedJ        9\edj         '    ' 
P>oni  the  definition  oi  6,  p  =  log  z  +  log  ^  and  hence 
(as  will  be  shown  presently)  for  large  z, 

(6)  H(z)  =  I  (log 2)2  +  log  z  [-  1  +  log(-  1)] 

+  log(l-.?)  +  <0)+{^+...  +  !^-hl(£)J  ;  lhr.e(.)  =  0 

where  c(^)  is  a  constant  depending  only  on  6.*  Since  6 
is  by  definition  a  periodic  function  of  z,  provided  z  moves 
along  a  straight  line  passing  through  the  origin,  the  same 
is  true  for  c{6}. 

It   remains  to  establish  equation   (G)  and   to  evaluate 
c(6)  and  the  as. 

*  Cf.  LiTTLEwoop,  loc.  cit.,  397  (13). 


ASYMPTOTIC   DEVELOPMENT 


9 


We  have  from  (5),  after  the  substitution 
p  =  log  z  -\-  log  ^, 
(7)  H(z^  =  (log  ^  4-  log  ^  +  1)  [log  2^  +  log  (^  -  1 )] 

_  llog  Qe  -  2)- (log  2  +  log(9  +  J)[log  z  4-  log  (60  -  1  j] 


+ 


(logg  +  logl^  +  iy 


+  (log  z  +  log  6)  [log  (ee  -  1)  -  1  -  log  ^  -  log  (1  -  6')] 

+.S'/  -f  n,(2:)  -  oxi?  + 1)  +  n,(j9)  -  n, (i). 

Consider  now 
We  may  write 


z     e^  y  -  1 


^os[(i-^^y-^)]-iog[(i-^i 


(-^) 


logfl 


gl+<!/\ 


j-log^l 


,1+ty         ^2(1+1?/)  ^3(1+22/) 


2  2:2 


8^^ 


+  — -\-- -^ 

Z  'Iz^  8^3 


+    •• 


-  ^  (cos  y  +  i  sin  ?/)  -  -f- j  (cos  2y  -\-i  sin  2  ?/ j  - 


+    f  cos^  —  ^  sin  y)  +  -(- j  (cos  2  ?/ —  z  sin  2  y) -f- 


-2z 


^  (sin  y)  +  i Q'  sin  2  y  +  iQ'  sin  3  y  -h  •  •  •  ] 


10  ASYMPTOTIC   DEVELOPMENT 

Moreover,  we  have  the  relation  * 

Sin  pX   n     _     IT  6;7+l_l 

^      e   Q  —  1         ^ 
Hence,  integrating  term  by  term, 

3WV2e3-l      3;^ 

We  have  a  right,  in  the  preceding,  to  integrate  term  by 
term,f  for,  put 


and  — "^ — =<l)(y^ 


0 


then,  (a)  when  ^  =  0,  Z^fnOy^  is  -  but  approaches  a 
definite    limit    -+(-)+(-)  +  •••as    y    goes    to    zero. 

In  determining  this  limit,  we  have  the  right  to  differen- 
tiate term  by  term  since  both  the  sine  series  and  the 
derived  cosine  series  are  uniformly  convergent  for  all  ?/'s, 

1. 


as 


(5)  When  y^^^  ^fnQy^  is  uniformly  convergent  for 

all   y\  and    I     I  <^(?/)  I  dy    is    convergent,  hence  we    may 

Jo 


'0 

integrate  term  by  term. 


*  BiERENS  DE  Haan,  TahU  of  Definite  Integrals^  264,  2. 
t  C  f .  Bkomwich,  Infinite  Series,  §  176  {A). 


ASYMPTOTIC   DEVELOPMENT 


11 


—  z 


Again,  consider 
We  may  write 


dy_ 


z    e 


2^y 


log 


,p+«2/ 


(-^) 


n-iogj 


!-»!/ 


(-^) 


log(l_^)_log(l 


Putting  —  =  ^,  and  proceeding  as  in  the  case  of  H^  (1), 


we  have 


a.(.)=-[.g^-i).l.g5^-l 


and  the  power  series  in  0  is  convergent  since  |^|  <  1, 
Consider 

''^^        ^  Jo        ^eP+1-^2/  — 2     e^'^y  —  l 

We  may  write 


losr 


,P+\+iy 


yP+\+ 


■■*]-i"g[(i- 


,p+i-t// 


,P+l-iy 


=  log    1 


'      )-log(l 


,P+1+»J/ 


,j^+l-?2/ 


+  2  2>. 


2;         1 
Puttinof  =  — -,  and  proceeding^  as  above,  we  have 

^a(,.i,=-i-[i(.!^;-.) 

\(]  V^le2  +  1      1\     1/1  VngS^i      IN  1 

22-1      27      3WV2e3_i      3;  J' 


+ 


'2.\e6. 


1    . 


and  the  power  series  in  -—  is  convergent  since 

eu 


<1. 


12 


ASYMPTOTIC   DEVELOPMENT 


In  the  same  way 

o/^       l_^rzfle-{-l     ,\  ,  Ifzyfle'+l      1 

^8VWV2e3-l     yy^     J 

and  liin  fl^  (x)  =  ^^g. 

Certain  further  reductions  may  be  made,  viz., 


-a.(i)  +  .;.,og^i-^J.i:^;^;(^). 


Also, 


and 


-  n.  (^  + 1)  +  fl.  (X)  +  r^i-'^n-r)^^ 


1    e"  4- 1  n  Y'. 


•^  2n     e«  -  1  Vg^y 
We  have,  then,  finally, 
(8)  If(z)  =  Klog  zy  +  log  2  [-  I  +  log  (-  1)] 


+ 

where 


^   + 


+  ^^  +  ^-  +  "^^^1;   l'me(2)  =  0 


<^)  =  -  iG^s  ^y  +  i  log  r^  - 1)  +  i  log  (.^  - 1) 


+  logn-l+log(-l)]-^-S:7 


1        e"  +  l 


1    2  71      e"  —  1 


6>^ 


ASYMPTOTIC   DEVELOPMENT  13 

and  ''^  =  i-.4±lf-«Y;A=l,2,...»-l. 

2*      2  k     e^-1  \zj 

^     1      e"  4- 1  feV'  . 
As  the  power  series  2y  ;" —      ^         ( -J   is  convergent,  evi- 
dently lim  €  (z)  =  0,  and  the  problem  of  §  3  is  completely 
solved. 


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